A Bayesian Semiparametric Quantile Regression Model for Longitudinal Data with Application to Insurance Company Costs

This article examines the average cost function for property and casualty insurers. Cost function describes the relationship between a firm’s minimum production cost and outputs. The comparison of cost functions could shed light on the relative cost efficiency of individual firms, which is of interest to many market participants and has been given extensive attention in the insurance industry. To identify and compare the cost function of insurers, common practice is to assume an identical functional form between cost and outputs and to rank insurers according to the center (usually mean) of the cost distribution. Such assumption could be misleading because insurers tend to adopt different technologies that are reflected by the cost function in their production process. We find that the cost distribution is skewed with a heavy tail. Therefore, the center-based comparison could lead to biased inference regarding an insurer’s efficiency in operation. To address these issues, we propose Bayesian semiparametric quantile regression approach to model the longitudinal data on production cost of insurance companies. Particularly we formulate the quantile regression using the asymmetric Laplace distribution; the effects of various firm characteristics on the quantiles of cost are specified as a linear function; the firm-specific relation between cost and multiple outputs is captured by a single-index function based on a spline basis where the coefficients are assumed to have a Dirichlet process prior. The single-index formulation with splines renders flexibility in modeling the nonlinear cost-output relation and the use of Dirichlet process leads to a natural clustering of insurers with similar cost efficiency. The method is applied to data on US property casualty insurers from the National Association of Insurance Commissioners (NAIC). The analysis of average cost at different quantiles indicates that better insights on efficiency are gained by comparing the whole cost distribution. A comparison of the model results with an external financial strength ratings for property casualty insurers provides interesting insights on which part of the cost distribution are perhaps weighted more by the rating agency